Summary of the reported records  and the calculated limits of Si and GaAs solar cells performances under the global AM1.5 spectrum.
In recent years there has been intense research work into the development of high efficiency solar cells relying on emerging novel materials and structures. All this has lead to a continuous record breaking of highest achievable efficiencies using different technologies. Since the first photovoltaic devices were developed the most prevalent concern is to hem in all sorts of efficiency losses preventing from reaching the physical limits [1-3]. To overcome this impediment, thorough investigations have been carried out to control and unearth their origins in order to identify potential efficiency advantages. Numerous thermodynamic approaches were employed to calculate solar cell efficiency limit, starting from the ideal Carnot engine to the latest detailed balance with its improved approach.
The aim of this chapter is to present a review of the techniques used to calculate the energy conversion efficiency limit for solar cells with detailed calculation using a number of numerical techniques. The study consists of analyzing the solar cell intrinsic losses; it is these intrinsic losses that set the limit of the efficiency for a solar energy converter. Several theoretical approaches were used in order to obtain the thermodynamic limit for energy conversion.
In the first place a solar cell could be considered as a simple energy converter (engine) able to produce an electrical work after the absorption of heat from the sun. In this fundamental vision the solar cell is represented by an ideally reversible Carnot heat engine in perfect contact with high temperature reservoir (the sun) and low temperature reservoir representing the ambient atmosphere. If the sun is at a temperature of
When the solar cell is supposed a blackbody converter absorbing radiation from the sun itself a blackbody, without creating entropy, we obtain an efficiency of about 93 % known as the Landsberg efficiency limit, which is slightly lower than Carnot efficiency.
Whilst a solar cell is assumed as an endoreversible system , the energy conversion efficiency is limited to 85.7% this figure is obtained where the sun is assumed fully surrounding the cell (maximum concentration). If we bear in mind that in a real situation the solar cell does not operate always in maximum concentration and the solid angle under which the cell sees the sun is in fact only a minute fraction of a hemisphere, the maximum efficiency is not larger than 12.79%, which is actually lower than most recently fabricated solar cells. However, we can conclude that solar cell is a quantum converter and cannot be treated as a simple solar radiation converter .
In the ideal model of a monochromatic cell incident photons are within a narrow interval of energy, while the cell luminescence outside this range is prevented. The overall resulting efficiency upper limit for an infinite number of monochromatic cells is 86.81% for fully concentrated sun radiation.
The ultimate efficiency of a single band gap
To represent a more realistic picture of a solar cell, three other fundamental factors should be taken into account, namely; the view factor of the sun seen from the solar cell position, the background radiation which could be represented as a blackbody at ambient temperature, and losses due to recombination, radiative and non-radiative.
In the detailed balance efficiency limit calculation first suggested in 1961 by Shockley and Queisser (SQ) in their seminal paper . It is assumed that illumination of semiconductor
The currently achieved short-circuit current densities for some solar cells are very close to predicted limits . Nevertheless, further gain in short-circuit current can therefore still be obtained, mainly by minimising the cell surface reflectivity, while increasing the thickness, so as to maximize the photons absorption. For thin film solar cells gain in photocurrent can be obtained by improving light trapping techniques to enhance the cell absorption.
Radiative recombination and the external fluorescence efficiency have a critical role to play, if the created photons are re-emitted out of the cell efficiently, which corresponds to low optical losses, the open circuit voltage and consequently the cell efficiency approach their limits . Concentrating solar radiation onto a solar cell improves remarkably its performance. Comparable effect could be obtained if the solar cell emission and acceptance angles were made equal.
2. Solar cell as a heat engine
2.1. Solar cell as a reversible heat engine
Thermodynamics has widely been used to estimate the efficiency limit of energy conversion process. The performance limit of solar cell is calculated either by thermodynamics or by detailed balance approaches. Regardless of the conversion mechanism in solar cells, an upper efficiency limit has been evaluated by considering only the balances for energy and entropy flux rates. As a first step the solar cell was represented by an ideally reversible Carnot heat engine in perfect contact with high temperature
for a reversible engine the total entropy is conserved, , then;
Hence the Carnot efficiency could be represented by:
This efficiency depends simply on the ratio of the converter temperature, which is equal to that of the surrounding heat sink, to the sun temperature. This efficiency is maximum (
Another way of calculating the efficiency of a reversible heat engine where the solar cell is assumed as a blackbody converter at a temperature
Under the reversibility condition the absorbed entropy from the sun
In accordance with the Stefan–Boltzmann law of black body, the absorbed heat flow from the sun is:
For a blackbody radiation, the absorbed density of entropy flow is:
The energy flow emitted by the converter at a temperature
And the emitted entropy flow is:
In this model the blackbody source (sun) surrounds entirely the converter at
And the transferred heat flow is:
The entropy-free, utilizable work flow is then:
Therefore the Landsberg efficiency can be deduced as:
The actual temperature of the converter
We arrive to a more general form of the Landsberg efficiency
Both forms of Landsberg efficiency (
In the Landsberg model the blackbody radiation law for the sun and the solar cell has been included, unlike the previous Carnot engine.
This figure represents an upper bound on solar energy conversion efficiency, particularly for solar cells which are primarily quantum converters absorbing only photons with energies higher or equal to their energy bandgap. On the other hand in the calculation of the absorbed solar radiation the converter was assumed fully surrounded by the source, corresponding to a solid angle of 4π.
Using the same approach it is possible to split the system into two subsystems each with its own efficiency; the Carnot engine that include the heat pump of the converter at
The ambient temperature is assumed equal to 300
The second part is composed of the sun as an isotropic blackbody at
The case of maximum concentration also corresponds to the schematic case where the sun is assumed surrounding entirely the converter as assumed in previous calculations.
Similar situation can be obtained if the solid angle through which the photons are escaping from the cell (emission angle) is limited to a narrow range around the sun. This can be achieved by hosing the cell in a cavity that limits the angle of the escaping photons.
The efficiency of this part of the system (isolated) is given by:
the resulting efficiency is simply the product:
This figure represents an overall efficiency of the entropy-free energy conversion by blackbody emitter-absorber combined with a Carnot engine. The temperature of the surrounding ambient
2.2. Solar cell as an endoreversible heat engine
A more realistic model has been introduced by De Vos et al.  in which only a part of the converter system is reversible, endoreversible system. An intermediate heat reservoir is inserted at the temperature of the converter
The net energy flow input to the converter, including the incident solar energy flow , the energy flow from the surrounding and the energy flow emitted by the converter is then:
The Müser engine efficiency (Carnot engine):
The converter temperature can be extracted from
And the solar efficiency is defined as ratio of the delivered work to the incident solar energy flux:
The maximum solar efficiency is then a function of two parameters; the Müser efficiency and the surrounding ambient temperature. From the 3d representation at figure 6 of the solar efficiency (
A general expression of solar efficiency of the Müser engine is obtained when the solar radiation concentration factor
Compared to Carnot efficiency engine the Müser engine efficiency, even when
If the ratio is fixed to 1/4, as in , which gives a good approximation for ambient temperature,
Hence, the corresponding
In the assumption of
3. Solar cell as a quantum converter
In a quantum converter the semiconductor energy band-gap, of which the cell is made, is the most important and critical factor controlling efficiency losses. Although what seems to be fundamental in a solar cell is the existence of two distinct levels and two selective contacts allowing the collection of photo-generated carriers .
Incident photons with energy higher than the energy gap can be absorbed, creating electron-hole pairs, while those with lower energy are not absorbed, either reflected or transmitted. The excess energy of the absorbed energy greater than the energy gap is dissipated in the process of electrons thermalisation, resulting in further loss of the absorbed energy. Besides, only the free energy (the Helmholtz potential) that is not associated with entropy can be extracted from the device, which is determined by the second law of thermodynamics.
3.2. Monochromatic solar cell
It is interesting to examine first an ideal monochromatic converter illuminated by photons within a narrow interval of energy around the bandgap . In the ideal case each absorbed photon yields an electron-hole pair. This cell also prevents the luminescent radiation of energy outside this range from escaping. According to the blackbody formula of the Plank distribution, the number of photons incident from the sun within an interval of frequency
The number of created electron-hole pairs, in the assumption that each absorbed photon yields an electron-hole pair, could be simply represented by: , where
This expression describes both the thermal radiation (for
This equation allows the definition of an equivalent cell temperature
When a solar cell formed by a juxtaposition of two semiconductors
At open circuit condition, the voltage
The density of work delivered to an external circuit (density of extracted electrical power)
The incoming energy flow from the sun can be written as:
The emitted energy density from the solar cell in a radiation form (radiative recombination) is:
The efficiency of this system is:
The work extracted from a monochromatic cell is similar to that extracted from a Carnot engine. The equivalent temperature of this converter is directly related to the operating voltage. At short-circuit condition it corresponds to the ambient temperature (
It is therefore possible to consider a monochromatic solar cell as reversible thermal engine (Carnot engine) operating between
We can see that an ideal monochromatic cell, which only allows radiative recombination, represents an ideal converter of heat into electrical energy.
In order to find the maximum efficiency of such a cell as a function of the monochromatic photon energy (
The monochromatic efficiency is considerable, particularly in the case of fully concentrated radiation (
To cover the whole solar energy spectrum an infinite number of monochromatic absorbers, each for a different photon energy interval, are needed. Each absorber would have its own Carnot engine and operate at its own optimal temperature, since for a given voltage the cell equivalent temperature depends on the photon energy (
3.3. Ultimate efficiency
The total number of photons of frequency greater than
This integral could be evaluated numerically.
In the assumption that each absorbed photon will produce a pair of electron-hole, the maximum output power density that could be delivered by a solar converter will be:
The solar cell is assumed entirely surrounded by the sun and maintained at
In accordance with the definition of the ultimate efficiency [6, 17], as the rate of the generated photon energy to the input energy density, its expression can be evaluated as a function of
This expression is plotted in figure 10, so the maximum efficiency is approximately 43.87% corresponding to
3.4. Detailed balance efficiency limit
The detailed balance limit efficiency for an ideal solar cell, consisting of single semiconducting absorber with energy band-gap
The model initially introduced by SQ  has been improved by a number of researchers, by first introducing a more exact form of radiative recombination. The radiative recombination rate is described using the generalised Planck radiation law introduced by Würfel , where the energy carried by emitted photons turn out to be the difference of electron-hole quasi Fermi levels. While for non-radiative recombination the released energy is recovered by other electrons, holes or phonons.
In the following sub-section the maximum achievable conversion efficiency of a single band-gap absorber material is determined.
3.4.1. Short-circuit current density (Jsc) calculation
Now we consider a more realistic situation of a solar cell, depicted in figure 3. Three factors will be taken into account, namely; the view factor of the sun seen from the solar cell, the background radiation is represented as a blackbody at ambient temperature
In steady state condition the current density
with reference to the solar cell configuration shown in figure 3,
The current density formula (43) can be rewritten in a more compact form as follows:
Under dark condition and zero bias the current density must be null, then:
Therefore the current density expression becomes:
From the above
In the ideal case the short-circuit current density depends only on the flux of impinging photons from the sun and the product
The currently achieved short-circuit current densities for some solar cells are very close to predicted limits. Nevertheless, further gain in short-circuit current can therefore still be obtained, mainly by minimising the cell surface reflectivity, while increasing its thickness, so as to maximize the photon absorption. For thin film solar cells the gain in
For instance crystalline silicon solar cells with an energy band-gap of 1.12 eV at 300K has already achieved a
3.4.2. Open-circuit voltage (Voc) calculation
At open circuit condition electron-hole pairs are continually created as a result of the photon flux absorption, the only mechanism to counter balance this non-equilibrium condition is recombination. Non-radiative recombination could be eliminated, whereas radiative recombination has a direct impact on the cell efficiency and particularly on the open circuit voltage. The radiative current as the rate of radiative emission increases exponentially with the bias subtracts from the current delivered to the load by the cell. At open circuit condition, external photon emission is part of a necessary and unavoidable equilibration process . The maximum attainable
In the case where
The current density expression becomes then:
The open circuit voltage can be deduced directly from this expression as:
The open circuit voltage is determined entirely by two factors; the concentration rate of solar radiation
Radiative recombination has a critical role to play, if the created photons are re-emitted out of the cell, which corresponds to low optical losses, the open circuit voltage and consequently the cell efficiency approach the SQ limit. Therefore the limiting factor for high
So dominant radiative recombination is required to reach high
If we define a maximum ideal open circuit voltage value
It is worth mentioning that (56) is not an exact evaluation of
From this figure one can say that taking
The other type of entropy loss degrading the open-circuit voltage is the photon entropy increase due to isotropic emission under direct sunlight. This entropy increase occurs because solar cells generally emit into 2π steradian, while the solid angle subtended by the sun is only 6.85×10−5 steradian.
The most common approach to addressing photon entropy is a concentrator system. If the concentration factor
With reference to table 1 we can clearly see that the record open circuit voltage under one-sun condition (
3.4.3. The efficiency calculation
Energy conversion efficiency
The maximum power is obtained numerically from (58) and the efficiency
For AM1.5G solar spectrum
Figure 13 illustrates efficiency against energy band-gap of a solar cell, using the AM1.5G spectrum and the blackbody spectrum at
In figure 14 the product of radiative recombination rate and the external fluorescence efficiency () is decreased from 1 to 10-3 to illustrate the effect radiative recombination and the external fluorescence on the cell efficiency. The maximum efficiency limit dropped from 34.42% to 28.58%.
Since the power output of the cell is determined by the product of the current and voltage, it is therefore imperative to understand what material properties (and solar cell geometries) boost these two parameters. Certainly, the short-circuit current in the solar cell is determined entirely by both the material absorption property and the effectiveness of photo-generated carriers collection at contacts. As previously mentioned (section 2.4.1), the manufactured solar cells with present technologies and materials have already achieved short-circuit currents close to predicted limits. Therefore the shortfall in efficiency could be attributed to the voltage. We show here that the key to reaching the highest possible voltages is first to have a recombination predominantly radiative with a maximal external emission of photons from the surface of the solar cell. Secondly we need a maximum solar concentration. The second condition could be achieved either by using sun concentrators, there are concentrators with concentration factor from ×2 to over ×1000  or by non-concentrating techniques with emission and acceptance angle limited to a narrow range around the sun [24-26].
At this level we can conclude that the efficiency limit of a single energy gap solar cell is bound by two intrinsic limitations; the first is the spectral mismatch with the solar spectrum which retains at least 50% of the available solar energy. The best known example of how to surmount such efficiency restraint is the use of tandem or stacked cells. This alternative will become increasingly feasible with the likely evolution of materials technology over the decades to 2020 .
The second intrinsic loss is due to the entropy associated with spontaneous emission. To overcame this limitation three conditions should be satisfied, that is: a) – prevailing radiative recombination (to eliminate the non-collected electro-hole pairs), b)-efficient external fluorescence (to maximise the external emission of photons from the solar cell) and c)-using concentrated sun light or restricting the emission and acceptance angle of the luminescent photons to a narrow range around the sun.
For single junction cell the record at present is 28.8% (GaAs)  compared to the SQ limit of 33.7% which is a significant accomplishment and little room has been left for improvement. Immense experimental research is now directed towards maximizing the external emission of photons from the solar cell. One way of getting beyond the SQ limit for a single junction is the use of concentrated radiation, the current record for concentrator cell is only 29.1% (GaAs) under 117 suns , this technology has a number of challenging problems (such as tracking and cooling systems) and there is still a long way to go. The same goal is accomplished by matching the angles under which light impinges from the sun and into which light is emitted from the solar cell. Recently it has been demonstrated that light trapping GaAs solar cell with limited emission angle efficiencies above 38% may be achievable with a single junction solar cell .
To overcome the restrictions of a single junction solar cell several directions were investigated during the last decades (i.e. hot carrier cells, carrier multiplication and down-conversion, impurity photovoltaic and multiband cells, thermophotovoltac and thermophotonic conversion…).
The most widely explored path has been tandem or stacked cells; they provide the best known example of how such high efficiency might be achieved. The present efficiency record for a triple junctions cell (InGaP/GaAs/InGaAs) is 37.9% compared to a predicted value of over 51% for an optimised set of three stacked cells [28, 29]. The major technological challenge with tandem solar cells is to find materials with the desired band gaps and right physical properties (i.e. lattice constant and thermal parameters). The ultimate efficiency target for this kind of configuration is 86.81% (for a set of an infinite number of stacked monochromatic cells under maximum solar concentration) which constitutes an arduous target, corresponding to an infinite number of stacked junctions radiated by a maximum solar concentration. The best performance that the present technology can offer is 44.4% using a triple junction GaInP/GaAs/GaInAs cell under 302 suns .
The author would like to thank Professor Helmaoui A. for his valuable and helpful discussions throughout the preparation of this work.